Two-Cocycles Give a Full Nonlinear Parameterization of the Simplest 3–3 Relation

Abstract

A parameterization of Grassmann-algebraic relations corresponding to the Pachner move 3-3 is proposed. In these relations, each 4-simplex is assigned a Grassmann weight depending on five anticommuting variables associated with its 3-faces. The weights are chosen to have the "simplest" form -a Grassmann-Gaussian exponent or its analogue (satisfying a similar system of differential equations). Our parameterization works for a Zariski open set of such relations, looks relevant from the algebraictopological viewpoint, and reveals intriguing nonlinear relations between objects associated with simplices of different dimensions.

Free Fermions on a Piecewise Linear Four-Manifold. I: Exotic Chain Complex

“…The existence of our algebraic realization of move 3-3 was first shown, and in a constructive way, in [5]. What was lacking in [5] was the natural nonlinear parameterization in terms of a 2-cocycle, found later in [6]. Once this parameterization is unveiled, further work leads -again quite naturally -to 3 Our analogue of functional integration will be a Berezin integral in anticommuting (Grassmann) variables, it will appear in the next part [8] of this work.…”

“…Our immediate argument in defense of chain complex (11) proposed in this paper is as follows: its construction can be shown to lead to the same, up to a 'gauge' freedom (see Subsection 5.3), pentachoron weights (16) as appear in [6] and [7] 4 . We will actually show in [8] that complex (11) provides an invariant of all Pachner moves, while papers [6] and [7] deal only with the move 3-3.…”

“…Note that, even in our discrete theory, some infinities that must be 'renormalized' do occur. Namely, they appear in disguise, as zeros 0 = ∞ −1 : we will get zero if we try to write, in a naïve way, a manifold invariant using the algebraic realization of Pachner move 3-3 from [6]. The 'exotic complex' proposed in the present paper can be treated as a tool for such renormalization, this latter consisting in replacing a vanishing determinant with a Reidemeister-type torsion.…”

“…We will actually show in [8] that complex (11) provides an invariant of all Pachner moves, while papers [6] and [7] deal only with the move 3-3.…”

“…In four dimensions, the 'central' move, in some informal sense, is move 3-3: it transforms a cluster of three 4-simplices situated around a 2-face into a cluster of three other 4-simplices, occupying the same place in the triangulation. An explicit algebraic realization of move 3-3 was given in papers [6,7] using Grassmann and Clifford algebras. Here algebraic realization means a formula whose structure corresponds naturally to the move 3-3, in such way that it gives strong support for further construction of a kind of topological quantum field theory.…”